Effects of Site-Specific Management on the Application of

1Corresponding author, Ph 509-335-8521, Fax 509-335-1173. [email protected]

Effects of Site-Specific Management on the Application of Agricultural Inputs

Center for Agricultural and Rural Development

Working Paper 96-WP 156

1996

David A. Hennessy*, Bruce A. Babcock**, and Timothy E. Fiez***

Department of Agricultural Economics*1

Washington State UniversityPullman, Washington

Department of Economics**

Iowa State UniversityAmes, Iowa

Department of Crop and Soil Sciences***

Washington State UniversityPullman, Washington


Abstract

Site-specific management of inputs in agricultural production is receiving increasing attention

because of new technologies and concerns about excessive input use. This paper provides a

microeconomic analysis of its implications. It shows that profit decreases with an increase in the

variability of input requirements, but that the input and production effects can be quite

complicated. The effects of moving from uncertainty about input requirements to variable

requirements are also identified. An empirical study of nitrogen fertilization suggests that site-

specific management may reduce input use substantially, but the production and profitability

impacts may not be large.

Key words: negative externality, site-specific management, uncertainty, variability.


Nitrogen and water share many common attributes. Both are essential for crop growth, and

both can be made available to the crop from either soil endowment or applied sources. Farmers

have control over applied water and nitrogen levels, but soil endowments are variable and may

be unknown. The use of both nutrients is of considerable interest to society because nitrogen

losses can contaminate water supplies and irrigation water often has a higher marginal value off

the farm than on it. A major concern among agronomists, environmentalists, and the water

industry, is the "excess" application of nitrogen by producers (Nielsen and Lee; Office of

Technology Assessment). Where the applied nutrient is distributed at a single rate across a field,

it has been shown that this "excess" could be explained by the interaction between the crop

production technology and the distribution of soil-stored nutrient (Babcock; Feinerman, Letey,

and Vaux).

New technologies now make it technically feasible to map soil nutrient status in small areas

of a field, and to use this information together with the guidance of global positioning satellites

to apply the appropriate input level at a given position (Brunoehler). This technology permits

the producer to move from a uniform, or single, application rate technology (SAR) to a site-

specific, or variable, application rate technology (VAR). Similar advances have occurred in

animal production. The availability of VAR gives rise to the question of its implications for

profit, output, and input decisions. Profit implications are of interest to producers and to the

providers of the agricultural machinery, computer hardware and software, and geographic

information systems (GIS) required to operationalize VAR. The output effect is of interest to the

food industry, while the effect on input use is of concern to nutrient suppliers, non-agricultural

water users and water quality/non-point pollution controllers.

Variability vs. Uncertainty

There is a clear differentiation between the concepts of variability and uncertainty.

Variability is where the soil-stored nutrient status is known, but not constant, across a field.

Uncertainty is where the status is not known. There are two distinct ways in which the soil-

2

stored nutrient can be variable. Spatial variability is where, at a particular point in time, the soil

status is not uniform across the fields' two dimensions. Temporal variability is where the soil

status changes over time, but at a particular point in time it is constant across the field.

Temporal variability is isomorphic to the spatial variability case when the decision-maker is

risk-neutral. Each year the producer identifies the soil nutrient status and distributes nutrient

accordingly. If one ignores time discounts, an integration over profit, or production, or input use

with respect to the a priori soil nutrient probability distribution makes the results under temporal

variability identical to the spatial variability case where the integration is with respect to the

mass distribution of soil nutrient across all field locations. Therefore, results pertaining to the

spatial variability case carry over to the temporal case with one qualification and one

interpretative exception. As qualification, assume that the distribution of soil stored nutrient is

independently and identically distributed from year to year. This is unlikely when nutrient

carryover occurs, and in rotations where different crops have different nutrient demands and

leave different soil residue profiles. In interpretation, instead of actual impacts of distributional

changes in soil nutrient status, we must talk of a priori expected impacts. This paper shall focus

on spatial rather than temporal impacts. Theoretical work on the economics of variability is

limited, but studies have been conducted by Katz, by Chavas and Larson, and by Chavas,

Kristjanson, and Matlon, while Niven as well as Fiez, Miller, and Pan, among others, have done

empirical work.

As with variability, there are two distinct ways in which soil nutrient status can be uncertain.

Spatial uncertainty is where, at a given time point, the status at any plot location is unknown.

The producer will apply a uniform rate of nutrient across the field. Here, if the cumulative

density function (cdf) of soil-stored nutrient is known with certainty, then producer risk

preferences do not matter because integration over the spatial dimensions maps, one for one,

each application rate onto a deterministic per plot production and profit. While production and

profit at a point may be stochastic, the law of large numbers ensures that whole field production

3

and profit are deterministic functions of the uniform nutrient application rate. For water,

uniform application under spatial uncertainty has been studied by Feinerman, Letey, and Vaux,

who found that differences in the shape of the production function can influence optimum water

application in non-intuitive ways.

Temporal uncertainty is where at any point in time the fields' nutrient status is invariant to

location, but total soil nutrient availability is unknown. There is a large literature devoted to the

study of anologues to the temporal case. Here, because of temporal non-substitutability of

utility, producer risk preferences matter. Some of the most recent and general work in this area

has been by Ormiston and Schlee (1992; 1993). Work of relevance to this study has been

conducted by Chiao and Gillingham who studied the impact of non-uniform distribution

methods on optimum application rates, Babcock and Blackmer who looked at the value of

resolving soil nitrogen uncertainty, and Babcock, Carriquery, and Stern who use a Bayesian

approach to estimate the value of incomplete resolution of soil nitrogen uncertainty.

When a producer adopts a VAR technology, the information environment he faces changes

from spatial uncertainty to spatial variability. Correspondingly, the problem faced changes from

choosing a SAR to choosing a function that maps known soil nutrient status onto application

rate. This change from uncertainty to variability is the focus of this study. To understand the

economic effects of the change we must understand economic decision-making under spatial

uncertainty and under spatial variability. As the literature review above suggests, the economic

consequences of uncertainty have received more attention than the consequences of variability.

In the next section of this paper we develop the theory of production under variability when soil-

stored and applied nutrients are additive. The following section considers the more general case.

Then the variability results are connected with the production under uncertainty results. And

finally we present an empirical study of variable rate nitrogen fertilizer application. In this way

insights will be developed on the effect on input use, production, and profit of moving from a

SAR to a VAR technology.

4

Q F ( x S x A) .(1)

P F ( x S x A) w x A,(2)

P F xAw 0 .(3)

Variability and an Additive Technology

Throughout this analysis is the random soil endowment of the nutrient at a point, isx x A

the level of nutrient applied by the decision-maker at a point, is the nonstochastic, per unitw

nutrient price, and is the nonstochastic output price. Land is assumed to be uniform inP

agronomic properties, except for the availability of the soil nutrient. Therefore, at all points on

the land surface the same production function applies. In this section we consider only the

increasing, concave, point-specific production function1

Later, we consider nonhomogeneous nutrients with a more general production function. We first

consider a change in the distribution function of . The initial cdf of is denoted by ,x S x S G 0(x S)

and the new cdf is denoted by . These distributions are supported in the interval [ ],G 1(x S) a , b

where must be non-negative and is assumed to be finite, but can be infinite. Wherea b

differentials are taken, they may be represented by any of the notations , , or d F

d x

F

xF x

where appropriate. If a variable, , is the th argument in the function, then may also bez i F i

used to denote the differentiation.

Returns over fertilizer costs for a unit area of land are

with profit maximizing first-order condition

By concavity, the second order condition is assured. Denote the optimum level of applied input

by .x

As previously discussed, temporal and spatial variability are isomorphic when decision-

makers are risk-neutral. We will apply the model to spatial variability but inferences can be

5

x A

x S

1 .(4)

w

b

a

[G 0(x S) G 1(x S) ] dx S,(5)

b

a

[G 0(x S) G 1(x S) ] d x S

drawn about the temporal case mutatis mutandis. The producer is assumed to know the

availability of soil nutrient at each point on the land surface. This knowledge may be completely

represented by the nutrient availability mapping , where y and z are spatial coordinates. x S(y , z )

At each spatial point, equation (3) is solved to give . A first degree stochasticallyx A(y , z )

dominating shift (FSD) in from the initial to the new distribution, cannot increase the totalx S

application of over all the field. In fact, if we assume that all of the initial and new massx A

distributions of are contained in the interior of [ ], then by differentiating (3) partiallyx S 0 , x

with respect to and , treating as an implicit function of , we getx A x S x A x S

Thus, there is a one for one reduction in the application rate as increases. In this case a FSDx S

in does not change total nutrient available at a point, so total production does not change. x S

Production will be constant across all points of the field. Profit will change due to savings in the

cost of nutrients. Profit increases by

where is the change operator, and can be shown, through an integration by parts, to be the

change in mean value of at a point. If some, but not all, of the new is contained in [ ],x S x S 0 , x

then some of the extra soil nutrient will not be compensated for by reductions in application

rates, output will increase, and profit will increase by a magnitude less than (5). Production will

6

t

a

[ G 0(x S) G 1(x S) ] d x S 0 for all t [a , b ] ,(6)

b

a

[G 0(x S) G 1(x S) ] dx S 0 .(7)

not be spatially uniform, being larger at points where the new is contained in the semi-openx S

interval ( ] than at points where the new is contained in [ ].x , b x S 0 , x

Next, consider a general mean preserving spread (mps) in the Rothschild and Stiglitz sense.

The mps requires that the following two conditions hold

If the mps occurs completely in the interval [ ], then the aggregate applied nutrient level,0 , x

output level, and profit level remain unaltered because of the complete substitution effect shown

by (4) above. If some of the mps occurs in the semi-open interval ( ], then the effect onx , b

aggregate applied nutrient level, output level, and profit level is ambiguous. The following can

be said, however. If the mps induces a decrease in mean , then total nutrient availability falls,x A

and the concavity of the production function will ensure that total output falls under a mps in . x S

However, if increases so that total nutrient availability increases then the two effects willx A

counteract to render ambiguity.

Because a second degree stochastically dominating shift (SSD) in can be decomposedx S

into a FSD and a mean preserving contraction (mpc) (Makowski; Hadar and Seo), we can infer

the effects of an SSD from the above analysis. If all of an SSD shift occurs in [ ], then0 , x

applied input use falls, profit rises according to (5) above, and output does not change. If some

of the SSD occurs in ( ], then the effects are ambiguous. If total nutrient availability rises,x , b

7

Q F ( x S, x A) ,(8)

P F 2 w 0 .(9)

then the mpc effect and production function concavity will ensure that total output rises. In this

case profit will also rise.

It should be noted that, except for the costs of collecting spatial information and of varying

application rates, if the original and new soil nutrient distributions are strictly positive only in the

interval [ ], then the only statistics relevant in determining profit and aggregate applied0 , x

input requirements are the original and shifted total soil nutrient levels. Further, in this case the

production level is independent of the soil nutrient distribution and shifts in distribution.

Because of complete substitutability between and , this result holds true regardless of thex S x A

number of factors in the production function. Elementary comparative statics on the first-order

conditions will verify this.2

Variability and a Nonadditive Technology

To this point we have considered only the situation where the applied nutrient and the soil-

stored nutrient are additive. This additivity assumption is not appropriate if the two nutrient

sources are not chemically identical, or if one source is more readily available to the plant.3 We

will next analyze the application decision when the two sources are not additive. This we do by

considering the following general, concave production function for a point on the landscape,

which gives the profit-maximizing first-order condition at a point,

It is assumed that is never so large as to make it uneconomic to apply nutrient at any point. x S

When is known to the producer, a soil nutrient contingent choice of , , can be madex S x A x A(x S)

to satisfy (9) for each value of . We now proposex S

PROPOSITION 1. A mps in the variability of decreases total field profit.x S

8

x A

x S

F 12

F 22

,(10)

P F [ x S, x A(x S) ] w x A(x S) .(11)

d 2

d x2S

P F 12

x A

x S

F 11 PF 11F 22 F

212

F 22

.

Proof. Solving (9), we get the input choice function . Using (9), the change in x A(x S) x A

with respect to that holds (9) constant isx S

Substituting into the profit function results inx A(x S)

Differentiating, and using the envelope theorem, gives . Differentiating againd / d x S P F 1

gives

Concavity of the production function ensures that the second derivative of with respect to x S

is negative. Therefore, by the Rothschild and Stiglitz generalization of Jensen's inequality, a

mps in will decrease profit.x S

An implication is that, given a fixed total amount of soil-stored nutrient, the greater the

locational variability the lower the value of land. We can also state

PROPOSITION 2. A mean preserving spread in the variability of increases (decreases)x S

total field production if is increasing (decreasing) in .F 1 F 2F 12/ F 22 x S

Proof. By Rothschild and Stiglitz, a mps increases (decreases) the expected value of a

convex (concave) function. We will identify the conditions under which production is convex

(concave) in . Differentiating the production function with respect to we getx S x S

. The result follows from signing a further differentiation, and using theF 1 F 2F 12/ F 22

Rothschild and Stiglitz result.

9

d 2F

d x2S

F 11F 22 F212

F 22

F 2

F222

2 F 12F 122 F 22F 112

F212F 222

F 22

.(12)

d 2x A

d x2S

1

F222

2 F 12F 122 F 22F 112

F212F 222

F 22

,(13)

d F 1 F 2

x A

x S

d x S

F 11F 22 F212

F 22

F 2

d F 12/ F 22

d x S

< 0 .

The resulting second derivative is

Due to the complexity of the expression, no global comparative static results should generally be

expected. A simpler result is

PROPOSITION 3. An increase in the variability of will increase (decrease) the use of totalx S

field if is decreasing (increasing) in .x A F 12/ F 22 x S

Proof. We will identify the conditions under which is convex (concave) in . Use thex A x S

first-order condition to get . From the first-order condition, the gradient isx A(x S)

. The result follows from differentiating this expression, and applying thed x A/ d x S F 12/ F 21

Rothschild and Stiglitz result on concave functions.

The resulting second derivative is

as has been reported previously by Katz. This expression is also rather difficult to sign. A

relationship between Propositions 2 and 3 is provided by the following corollary,

COROLLARY 1. An increase in the variability of that leads to a decrease in the use ofx S

total field always leads to a decrease in total field output.x A

Proof. From Proposition 2, an increase in the variability of decreases output ifx S

10

x S) ] [x S, x A(x S) ] (x S x S) [x S, x A(x S) ]1

2(x S x S)2 [x(14)

The first right hand expression is negative due to the concavity of the production function. The

positivity of the full derivative in the second right hand expression is a necessary and sufficient

condition for to decrease with a mps in .x A x S

The result is due to the direct effect of a mps on yield and the reduction in total field bothx A

acting in the same direction. Having developed the economic implications of variability, in the

next section we will connect these results to the literature on production under uncertainty. This

connection will allow us to study the transition from a SAR technology to a VAR technology.

Moving from Uncertainty to Variability

From Propositions 2 and 3, it is clear that unambiguously signing the effects of spatial

variability on output and application rates may not always be possible for the general production

function. In this section we will show that, perhaps somewhat counterintuitively, when one

moves from uncertainty to variability with the same distribution of the variable by acquiring

information about the distribution of , the effect on production and input use may be easier tox S

sign. Let be the optimal SAR, let be the optimal SAR when has mean and zerox A x A (x S) x S x S

variance, and let be the variance of the soil nitrogen distribution. The results obtained below2S

are based on second-order approximations which have low errors for tight distributions (low

variances) of .x S

PROPOSITION 4. For tight distributions of , the change in expected profits, , fromx S E [ ]

acquiring ex-ante site-specific information is approximately

.2S

P

2

F212

F 22

P

2[x A x A (x S) ]2F 22

Proof. We take the difference of second-order approximations of profit when is variablex S

but known and expected profits when is uncertain. First, take a second-order x S

Taylor series expansion of profit around the mean, :x S

11

E { [x S, x A(x S) ]} [x S, x A(x S) ]1

2P

2S

F 11F 22 F212

F 22

.(15)

) [x S, x A (x S) ] (x S x S) 1 [x A x A (x S) ] 21

2(x S

(x S x S) [x A x A (x s) ] 121

2[x A x A (x S) ]2

22.

(16)

E [ (x S, x A ) ] [x S, x A (x S) ]P

2{

2SF 11 [x A x A (x S) ]2F 22} .(17)

E { [x S, x A(x S) ]} E [ (x S, x A ) ]2S

P

2

F212

F 22

P

2[x A x A (x(18)

where the prime indicates a complete first derivative with respect to (including the effectx S

through ), and the double prime indicates a complete second derivative with respect to . x A x S

Noting that , and taking expectations of both sides of (14)[x S, x A(x S) ] P (F 11F 22 F212) / F 22

results in

Now we need an expression for expected profits under uncertainty. Profit at any location is a

function of two variables: and . Take a Taylor-series expansion of profit at any spatialx S x A

location around the point and the input choice , where denotes the optimalx S x A (x S) x A (x S)

input level when there is no spatial variability:

Taking expectations and substituting the appropriate expressions for the derivatives gives

Subtracting (17) from (15), and noting that the choice under variability equals the choicex A(x S)

under uncertainty, results in x A (x S)

which completes the proof.

COROLLARY 2. An increase in the variability of increases .x S E [ ]

12

d E [ ]

d2S

P

2

F212

F 22

P [x A x A (x S) ] F 22

x A

2S

.(19)

x A(x S) x A(x S) (x S x S)d x A

d x S

(x S x S)2

2

d 2x A

d x2S

.(20)

d E [x A(x S) ]

d2S

1

2 F222

2 F 12F 122 F 22F 112

F212F 222

F 22

.(21)

Proof. Differentiating with respect to results in E [ ]2S

When ; then , and . When ; thend x A / d2S > 0 x A > x A(x S) d E [ ] / d

2S > 0 x A /

2S < 0

, and .x A < x A(x S) d E [ ] / d2S > 0

Thus, as one would expect, the value of moving to known variability increases as spatial

variability increases. This last proof raises the question as to how increases in spatial variability

affect the change in mean input use as one moves from uncertainty to known variability.

PROPOSITION 5. For tight distributions of , an increase in spatial variability changes thex S

difference in expected input use, , by approximatelyE [x A]

.1

2 F222

2 F 12F 122 F 22F 112

F222F 112

F 22 [x A x A (x S) ] F 222

F212F 222

F 22

Proof. Again we will take the difference in second-order approximations. A second-order

approximation around obtainsx S

Taking expectations of both sides, differentiating with respect to , and substituting in for the2S

expression results in d 2x A/ d x2S

Now we need to approximate the difference between the marginal product of applied inputs and

the price ratio at a given location in the field,

13

S, x A )w

PF 2[x S, x A (x S) ]

w

P(x S x S) F 12 [x A x A (x S)

1

2(x S x S)2F 112 (x S x S) [x A x A (x s) ] F 122

1

2[x A x A (x S) ]2F

(22)

E [F 2(x S, x A ) ]w

P0

S, x A (x S) ]w

P[x A x A (x S) ] F 22

1

2

2SF 112

1

2[x A x A (x S

(23)

d x A

d2S

1

2

F 112

F 22 [x A x A (x S) ] F 222

.(24)

A] 1

2 F222

2 F 12F 122 F 22F 112

F222F 112

F 22 [x A x A (x S) ] F 222

F21

(25)

d E [x A]

d2S

1

2 F222

2 F 12F 122

F212F 222

F 22

.(26)

Taking the expectation of both sides and using the first-order condition results in

Differentiating both sides of (23) with respect to and solving for results in2S d x A / d

2S

Take the difference between (21) and (24) to obtain


Note that when , that is, our departure point is where there is no uncertainty,x A x A (x S)

then (25) becomes

The conventional wisdom is that (26) is often negative, particularly when is small relative tow

the average value of . That is, adoption of site-specific farming practices should decreasex A

input use. is the Rothschild and Stiglitz concavity condition applied to the first-orderF 112

14

S, x A(x S) ] F [x S, x A(x S) ]

2S

2F 11 2 F 12

d x A

d x S

F 22

d x A

d x S

2

F 2(27)

S, x A ) ] F [x S, x A (x S) ] [x A x A (x S) ] F 2

2S

2F 11

[x A x A (x S)

2(28)

condition. This is removed from the variability effect to purge it of the impact of uncertainty,

and leave only technical substitution impacts associated with certain variability. If , as isF 12 < 0

likely in our case, then the local effect of moving from uncertainty to variability is to increase

average input use if and , while the effect is to decrease average input use ifF 122 < 0 F 222 > 0

and . The effect is indeterminate if these two third derivatives have the sameF 122 > 0 F 222 < 0

sign.

Having approximated the input effects, it should be possible to identify the production

effects of site-specific information.

PROPOSITION 6. For tight distributions of , an increase in spatial variability changes thex S

difference in expected yield, , from acquisition ofE [F (x S, x A ) ]

information by approximately

.F

212

2 F 22

F 2

2 F222

2 F 12F 122 F 22F 112

F212F 222

F 22

{ F 2 [x A x A (x S) ] F 22}x A

2S

Proof. Using equation (12), the expectation of a second-order expansion of yield under

variability is

And the expectation of a second-order expansion of yield under uncertainty is

Taking the difference between (27) and (28), and differentiating with respect to gives2S

15

d E [F [x S, x A ]

d2S

F212

2 F 22

F 2

2 F222

2 F 12F 122 F 22F 112

F212F 222

F 22

{ F 2 [x A x A (x S) ] F 22}d x A

d2S

,

(29)

d E [F (x S, x A ) ]

d2S

F212

2 F 22

F 2

2 F222

2 F 12F 122

F212F 222

F 22

(30)

Pd E [F (x S, x A ) ]

d2S

d E [ ]

d2S

wd E [x A]

d2S

.(31)


When , (24) and (29) givex A x A (x S)

Under this condition, we can use propositions 4, 5, and 6 to write

That is, the change in expected profit due to the shift from uncertainty to variability added to the

change in expected cost due to the shift equals the change in expected revenue due to the shift.

A logical inference from (31) is

COROLLARY 3. A shift from uncertainty to variability in that leads to an increase inx S

mean use of always leads to an increase in mean output.x A

Proof. The proof follows from equation (31) and the fact that expected profit must increase

with the shift.

This result arises because site-specific information improves the efficiency of nitrogen use.

If nitrogen is used more efficiently and if more of it is used, then mean production must increase.

Having developed the theory, in the next section we will apply it to a nitrogen use problem in the

Palouse area of eastern Washington.

16

Data and Dynamic Considerations

The Palouse region of eastern Washington state has a distinctive topography, consisting of

fertile rolling hills. Hill slope averages about 13%, and Mulla et al. have concluded that soil

fertility varies considerably within a field. This suggests a need for good information on

fertility. The principal rotation in the eastern Palouse consists of winter wheat followed by

spring barley and then a legume. The legume is either dried pea or spring lentil, and wheat is the

most profitable crop. As wheat follows the legume and legumes endow the soil with high but

variable amounts of nitrogen, the region may be particularly well suited for variable rate

technology.

Fiez, Miller, and Pan conducted a series of nitrogen experiments on white winter wheat at

two silt loam soil type locations in eastern Whitman county on the border with Idaho. The

variety chosen was Madsen, commmon in the area. At both locations (Pullman and Farmington)

the experiments were carried out over the successive years 1990 and 1991. The previous crop

was lentils at Farmington and peas at Pullman. This previous cropping pattern is true in both

years because the fields at each location were not the same in the two years. In 1990 five

different applications rates (0, 50, 75, 100, and 125 lb/acre) of aqua ammonia were applied at

planting. In 1991 a sixth application rate (25 lb/acre) was added. Also considered were each of

the four basic landscape positions (south backslope, shoulder, north backslope, and footslope).

The rainfall was average for the region in both years (20.4 inches from September to September

in 1989-1990, and 20.8 inches in the following year), and irrigation was not used.

For each block of replications (five application rates in 1989-1990, and six application rates

in the following year), preplant inorganic residual soil nitrogen was measured in a 60 inch soil

profile from the surface. Nitrogen mineralization was also imputed from readings of soil organic

matter. These two sources of nitrogen were summed to give a measure of the preplant soil

nitrogen status. A total of 340 plot yields were recorded, twelve less than the number of planted

plots. These twelve 1991 Farmington shoulder slope plots were eliminated because winterkill

17

xt 1S (x

tS x

tA) ,(32)

Max

xt

A, xt 1A , . . . i 0

i [P F ( xt iS x

t iA ) w x

t iA ]

(33)

PF (x

tS x

tA)

x A

w w(34)

severely reduced the plant stand.

Farm level white wheat price was assumed to be $3.50 bu/acre, and the price of applied

nitrogen was assumed to be $0.31/lb (Painter, Hinman, and Burns). However, not all nitrogen

applied in a year is actually taken up in that year. Depending on weather conditions and the

nature of the agriculture, a non-negligible fraction may be carried over as soil nitrogen to the

following crop year. In a study of sorghum production in Australia's Northern territory,

Kennedy et al. set the range of carryover at between 20% and 40%. For Iowa, Fuller estimated a

carryover for continous corn of about 32%. This carryover should be accommodated in any

static optimization model. We shall adopt the approach of Kennedy, outlined below.

Let carryover be a constant proportion of the sum of applied and soil nitrogen,

where is the fractional carryover and the superscript denotes the annual time period. Then the

producer faces the dynamic programming problem

subject to the carryover equation, nonnegative nitrogen applications, and an initial soil nitrogen

endowment. Here, is the annual discount factor. Assuming a concave production function,

Kennedy developed an optimal steady state decision rule. That rule is

to apply at time the nitrogen level which solves t

Given constant prices and a concave production function, carryover equation (32) ensures that

soil and applied nitrogen levels will converge to time invariant steady state levels. In equation

18

A) b 0 b Sx S b Ax A b SSx2S b SAx Sx A b AAx

2A b SSSx

3S b SS

b SAAx Sx2A b AAAx

3A c 1FA c 2BS c 3FS c 4RT c 5YR ,

(35)

F (x S, x A) d 0 d 1(x S x A) d 2(x S x A)2 d 3(x S x A)3

c 1FA c 2BS c 3FS c 4RT c 5YR

(36)

(34), note that discounted marginal value of product is set to a fraction of input price rather than

the full input price. This effective input price, , is decreasing in the discount rate and(1 ) w

in the carryover fraction.

Estimation and Results

To capture the production and input effects developed in the theory section above, it is

necessary to estimate a production function that is flexible to the third order. We chose the

general cubic production function,

where is a dummy variable equal to one when the observation is at the Farmington site. ,FA BS

, are dummy variables equal to one when the observation is a south backslope, aFS RT

footslope, and a north backslope, respectively. is a dummy variable set equal to one in theYR

1990-1991 crop year. Yield is measured in bu/acre, and nitrogen in lb/acre. Regression results

are presented in table 1. The coefficients of the powers of are all strongly significant. Sox S

also are the location and year coefficients, though the site coefficient is not. The terms with

applied nitrogen in them are not individually significant. An F test was run to test for the

collective significance of these terms. The result is presented in table 2, and it was found that

they were collectively significant at the 1% level. The and coefficients are negative,b SS b AA

and so are consistent with concavity. The coefficient is negative, which is consistent withb SA

the two sources of nitrogen being substitutes. The perfect substitutability specification

was tested for (see table 2), and rejected at the 1% level.

The coefficient is positive, suggesting that a more uncertain distribution of soilb SSA

19

nitrogen would increase demand for applied nitrogen. This is consistent with Babcock's (1992)

suggestion that nitrogen is applied as insurance against the possibility of low soil nitrogen. The

model was also tested for the hypothesis that the coefficients of own power terms are equal; i.e.,

imposing the restrictions that , , and . This is a less stringentb S b A b SS b AA b SSS b AAA

version of the perfect substitution hypothesis. The test result is presented in table 2, and the

restrictions were also rejected at the 1% level.

Because the comparative static conditions arrived at in the theory sections are not simple, it

was not possible to construct confidence intervals around point estimations to test statistically

for production and input use impacts of increases in variability and of moving from uncertainty

to variability. Point estimates are provided in table 3. These estimates are evaluated at the mean

level of soil nitrogen over all plots, 198.6 lb/acre, and the expected profit maximizing level of

applied nitrogen when the white wheat price is $3.50/bu and the effective nitrogen price is

$0.22/lb, This profit maximizing level of nitrogen is 63 lb/acre. Applying equation (34), if the

discount rate, , equals 0.93, and the actual nitrogen price is $0.31/lb, then an effective nitrogen

price of $0.22/lb is consistent with a carryover fraction of 0.312, a reasonable proportion.4

From the second order conditions, we conclude that the production function is concave. To

preserve space, the three conditions are not presented in table 3. When is endogenized due tox A

a variable but certain distribution of , then the production function is concave with secondx S

derivative equal to -0.00822 at the point of evaluation. Therefore, a mps in the variability of x S

is expected to decrease mean production, in accordance with Proposition 2. With a variable but

certain distribution of , then is concave with second derivative equal to -0.02719 at thex S x A(x S)

point of evaluation, indicating that a mps in the variability of should decrease mean input use,x S

in accordance with Proposition 3.

As the last three results in the table are the evaluations of derivatives with respect to , to2S

fully interpret their implications it is necessary to have knowledge about the magnitudes of . 2S

The means and variances of for the two sites and four topographical locations are presented inx S

20

table 4. Standard deviations are about 25 lb/acre, except for two sites where some very high

outliers caused high standard deviations. For the purpose of interpreting table 3, we will assume

a standard deviation of 35 lb/acre. In this case mean profit increases by a meagre 0.00015(35)2 =

$0.184/acre, hardly enough to sustain a management intensive innovation. The effect on mean

input use is more substantial, however, at - 0.01891(35)2 = - 23 lb/acre. But this reduction in

input use does not seem to affect mean output much because it appears to fall by only about -

0.00121(35)2 = - 1.48 bu/acre. It would appear that, while the input use impacts may be

significant, the profit implications might not be. As the above results are just approximations

and the model is in place to provide more exact results, next we will provide simulations to

confirm our inferences.

First, we will evaluate optimal decisions for the expected profit maximizer without access to

site-specific soil nitrogen readings. We will then optimize when site-specific information is

available, and compare the two sets of results. We assume that our expected profit maximizer

knows the spatial distribution of soil nitrogen for each of the four slopes and two fields

considered in the experiments. We estimate optimal nitrogen application rates for each of the

slopes and fields. From Painter, Hinman, and Burns, variable costs other than nitrogen were

assumed to be $94.23/acre, and fixed costs were assumed to be $121.26/acre.

Table 5 presents the results for the expected profit maximizer without access to site-specific

information. At each site the first-order condition was solved for the value of that resulted inx A

the highest average profit at that site. At this rate, average yield and profit were calculated. As

shown in table 4, Farmington shoulder and Pullman north backslope locations have sample

standard deviations of soil nitrogen that are much greater than the other locations. These large

values presented us with difficulties because the regularity conditions of the production function

were violated at some soil nitrogen levels. For this reason, the standard deviations at these plots

were reduced by 25 lb/acre when solving for both SAR and VAR nitrogen application rates.

They were still almost twice as high as standard deviations for other plots.

21

The mean application rate under SAR technology, giving each slope X location combination

equal weight, is 58 lb/acre. Typical rates in this region are 90-100 lb/acre, so our average rate is

somewhat low. Expected yield under the SAR technology varies from 99.4 bu/acre on the

Pullman shoulder sites down to 82.2 bu/acre on the Pullman north backslope sites, where soil

nitrogen was high enough to make it unprofitable to apply any fertilizer (see table 4). Mean

profit for the SAR technology ranges from $60/acre to $120/acre.

To calculate optimal application rates under VAR, the first two moments of the soil nitrogen

distribution at each of the eight slope X location combinations was used to given the moments of

a random normal variate. Then the first-order condition (9) is solved for each of 3,000 random

draws from this distribution. Because the production function is cubic, there are two nitrogen

application rate solutions for each draw, but only one is relevant. Profit and output were

calculated for the optimal nitrogen application for each draw, and then profit, output, and input

use were averaged over all draws. The results are presented in parentheses in table 5.

The average nitrogen application rate under VAR is less than under SAR for all locations

except for the Pullman north backslope, where the optimal rate is zero. The average reduction

on the other seven sites is 17 lb/acre, which represents a 25.5% reduction. Overall, under the

VAR technology average nitrogen application is 44.9 lb/acre, a 22.6% reduction. The largest

decrease is on the shoulder slopes at Farmington where soil nitrogen variance is high. At this

Farmington site, much of the nitrogen is applied under SAR as assurance against being caught

short of nitrogen. Under VAR, this assurance is no longer necessary.

Average yield under VAR is 87.1 bu/acre, down 0.7 bu/acre from the SAR scenario.

Average profit is $79.91/acre, up only $0.37/acre from the SAR scenario. There would appear to

be little economic incentive for producers in this area to adopt this technology for nitrogen

application.

Summary and Conclusions

This paper has demonstrated the complexity of the impacts of site-specific information on

22

decision-making. While information has positive value, its effects on production and input use

are not clear. In practice, it seems likely that moving from input use under uncertainty to input

use under variability will decrease mean input use because uncertainty may be causing privately

excessive levels of input use as insurance against the possibility of low yields. Empirical results

provide evidence for these conclusions.

However, our results suggest a low value for site-specific information, so there may be little

incentive to adopt the technology. As reported by Brunoehler, Lowenberg-DeBoer estimates the

sampling and mapping costs of site-specific management at about $7.25/acre. Lowenberg-

DeBoer goes on to suggest that it would be difficult to cover this cost through reducing average

input levels. In general, the value of site-specific management may vary from site to site, crop to

crop, and input to input. For example, the variable application of expensive patented pesticides

may be quite profitable. Because we have controlled for year, our results do not suggest that soil

testing is irrelevant. As Babcock and Blackmer have demonstrated, there may be considerable

demand for the resolution of temporal uncertainty.

The results suggest interesting policy implications for governments. Suppose that

uncertainty is causing privately excessive applications of inputs that cause negative externalities,

and the government owns global positioning infrastructure. If this infrastructure is not

congested, then its services are public goods. If the fees for positioning services make site-

specific management unprofitable, then to reduce the magnitudes of the negative externalities,

the government might consider refraining from charging fees.

23

Footnotes

1. If is considered not as a soil nutrient endowment but as an index of technicalx S

productivity instead, then variability in can be considered as heterogeneity in technology. Forx S

nitrogen, the effect of the introduction of GIS on nitrogen application when there are

heterogeneous soil conditions in a field has been studied by Niven (1994). The index of

technical productivity interpretation can be applied to all production functions considered in this

paper.

2. The production function , where and are chosen inputs, is of littleQ F (x S x 1, x 2) x 1 x 2

interest when is variable but known. If does not exceed , then supplements tox S x S x x 1 x S

give always, and the comparative statics matrix has a zero determinant. That is, isx x 2

unaffected by the distribution of .x S

3. This is particularly true when we generalize and consider, for example, substitutability

between homegrown fodder and purchased, quality-controlled, concentrate animal feed.

4. Because our formula for computing the carryover fraction, equation (34), is predicated on

the assumption of perfect substitutability, an assumption that was rejected, the calculated

carryover fractions can only be considered to be rough approximations.

24

References

Babcock, Bruce A. "The Effects of Uncertainty on Optimal Nitrogen Applications." Rev.Agr. Econ. 14(July 1992):271-80.

Babcock, Bruce A., and Alfred M. Blackmer. "The Value of Reducing Temporal InputNonuniformities." J. Agr. Resour. Econ. 17(December 1992):335-47.

Babcock, Bruce A., Alicia L. Carriquery, and Hal. S. Stern. "Evaluation of Soil TestInformation in Agricultural Decision Making." Applied Statistics, in press.

Brunoehler, Ron. "Honing in on Site-Specific Production." AgriFinance, the journal ofagricultural professionals, June/July 1995, pp 28-31.

Chavas, Jean-Paul, and Bruce A. Larson. "Economic Behavior Under TemporalUncertainty." South. Econ. J. 61(October 1994):465-77.

Chavas, Jean-Paul, Patricia M. Kristjanson, and Peter Matlon. "On the Role of Informationin Decision Making: The Case of Sorghum Yield in Burkino Faso." J Dev. Econ.35(April 1991):261-80.

Chiao, Yen-Shong, and Allan Gillingham. "The Value of Stabilizing Fertilizer under Carry-Over Conditions." Amer. J. Agr. Econ. 71(May 1989):352-62.

Feinerman, Eli, J. Letey, and H.J Vaux Jr. "The Economics of Irrigation with NonuniformInfiltration." Water Resour. Res. 19(December 1983):1410-14.

Fiez, Timothy E., Baird C. Miller, and William L. Pan. "Assessment of Spatially VariableNitrogen Fertilizer Management in Winter Wheat." J. Prod. Agr. 7(1 1994):86-93.

Fuller, Wayne A. "Stochastic Fertilizer Production Functions for Continuous Corn." J. Farm Econ. 47(1965):105-19.

Hadar, Josef, and Tae Kun Seo. "The Effects of Shifts in a Return Distribution on OptimalPortfolios." Int. Econ. Rev. 31(August 1990):721-736.

Katz, Eliakim. "On Variability under Certainty." Eur. Econ. Rev. 26(October-November1984):109-13.

Kennedy, John O.S. "Rules for Optimal Fertilizer Carryover: An Alternative Explanation."Rev. Market. Agr. Econ. 54(August 1986):3-10.

Kennedy, John O.S., Whan, I.F., Jackson, R., and John L. Dillon. "Optimal FertilizerCarryover and Crop Recycling Policies for a Tropical Grain Crop." Aust. J. Agr. Econ.17(2 1973):104-13.

Makowski, Armand M. "On an Elementary Characterization of the Increasing ConvexOrdering, with an Application." J. Appl. Prob. 31(1994):834-840.

Mulla, David J., A.U. Bhatti, Max W. Hammond, and J.B. Benson. "A Comparison of

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Wheat Yield and Quality under Uniform vs. Spatially Variable Fertilizer Management."Agric. Ecosyst. Environ. 38(1992):301-11.

Nielsen, E., and Linda Lee. "The Magnitude and Costs of Groundwater Contamination fromAgricultural Chemicals: A National Perspective." Washington D.C: United StatesDepartment of Agriculture, ECRS, AER-576, October 1987.

Niven, Victoria. "Agricultural Input Response to Soil Spatial Variability." unpublishedMasters thesis, Iowa State University, Ames IA., 1994.

Office of Technology Assessment. Protecting the Nation's Groundwater fromContamination. Washington D.C: United States Government Printing Office, 1984.

Ormiston, Michael B., and Edward E. Schlee. "Necessary Conditions for ComparativeStatics under Uncertainty." Econ. Let. 40(December 1992):429-34.

Ormiston, Michael B., and Edward E. Schlee. "Comparative Statics under Uncertainty for aClass of Economic Agents." J. Econ. Theory 61(December 1993):412-22.

Painter, Kathleen, Herbert R. Hinman, and John Burns. 1995 Crop Rotation Budgets forEastern Whitman County, Washington. Extension Bulletin 1437, Cooperative ExtensionService, Washington State University, Pullman, WA.

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26

Table 1. Econometric Estimates of the Cubic Production Function Parameters

Parameter Estimate t-statistic

-164.27 - 6.78b 0

2.85283 9.83b S

0.312845 1.43b A

- 0.00979577 - 8.64b SS

- 0.00161579 - 1.15b SA

- 0.00105106 - 0.47b AA

0.00001058 7.44b SSS

0.00000207 0.85b SSA

0.00000485 0.88b SAA

0.000000235 - 0.02b AAA

- 0.81479 - 0.82c 1

- 8.32977 - 5.76c 2

- 9.90051 - 7.20c 3

- 13.2722 - 8.64c 4

- 2.44571 - 2.05c 5

Table 2. Tests for Nested Specifications of the Cubic Production Function

Test F-value Degrees of Freedom andSignificance

Collective significance ofterms with in themx A

5.819 6, 325 1%

Perfect substitutabilitybetween and x S x A

25.81 6, 325 1%

Coefficients of own powersare set equal

28.01 3, 325 1%

Cubic terms are set equal tozero

14.10 4, 325 1%

27

Table 3. Point Estimates of Comparative Static Effects

Comparative Static Relevant Proposition Estimate Conclusion

Concavity of in itsF (x S, x A)

arguments

Proposition 1 ** Concave

Curvature of in Proposition 2 - 0.00822 mps decreases meanF [x S, x A(x S) ] x S

production

Curvature of in Proposition 3 - 0.02719 mps decreases meanx A(x S) x S

input use

Approximate value of

information about x S

Proposition 4 0.00015 low increase in profit

Input effect of information on Proposition 5 - 0.01891 significant decreasex S

in mean input use

Approximate production effect of

information on x S

Proposition 6 - 0.00121 low decrease in

production

Table 4. Moments of the Distribution of Soil Nitrogen

Site TopographicalLocation

Meanlb/acre

Standard deviationlb/acre

Farmington South backslope 176.1 33.53

Footslope 200.8 24.36

North backslope 186.1 22.66

Shoulder 216.3 86.25

Pullman South backslope 168.8 19.46

Footslope 187.3 25.67

North backslope 253.7 81.06

Shoulder 193.6 19.15

28

Table 5. Profit, Yield, and Nitrogen Application by Site, Location, and Technology (Variable Rate Technology Results are in Parentheses)

Nitrogen @$0.22/lb

Farmington Southbackslope

59.54 (59.64) 83.00 (82.14) 71.9 (57.8)

Footslope 85.43 (85.56) 89.23 (88.32) 53.5 (39.6)

Northbackslope

61.76 (62.50) 83.14 (82.66) 64.4 (53.4)

Shoulder 98.76 (99.22) 94.65 (91.70) 79.3 (30.3)

Pullman Southbackslope

62.47 (63.48) 83.85 (83.84) 71.8 (67.1)

Footslope 75.79 (75.83) 87.21 (86.36) 64.6 (50.9)

Northbackslope

72.46 (72.90) 82.24 (83.19) 0 (13.1)

Shoulder 120.03 (120.11) 99.42 (98.70) 58.6 (46.8)

Documents

Effects of Site-Specific Management on the Application of